The next generation wireless communication systems demand for even higher spectral efficiencies to accommodate the higher throughput requirements within the limited frequency bands. Multiple antenna (MIMO) systems and in particular closed loop transmission technologies such as beamforming and precoding have been vastly considered to improve the spectral efficiency. In MIMO precoding schemes, the transmitted data is divided into several independent streams where each streams is individually precoded and all precoded streams are superimposed before transmission. The number of transmitted streams is called transmission rank. The transmission rank can be optimally chosen for a given channel realization by considering the transmit power and the overall channel statistics. For each stream the precoder is a beamforming vector which consists of a set of weights, one for each antenna, that are multiplied with the transmitted symbol prior to the transmission.
The precoder of rank r is then a matrix consists of r beamforming vectors as its columns. Generally, the average transmit power is divided equally between all streams, thus, the norm of all beamforming vectors are equal and normalized to one. Furthermore, due to the superposition of the transmitted streams, the beamforming vectors are chosen to be orthogonal for simplicity of the decoding. As a result, the precoder is generally a semi-unitary matrix (Vn×r, r≦n is called semi-unitary if and only if VHV=1).
In codebook based precoding strategies, a predetermined codebook will be made available to the transmitter, i.e., Base Station (BS), and all receivers, i.e., Mobile Stations (MS). The receiver will then choose a precoder from the codebook which maximizes its performance (e.g. its data rate) and feeds back the precoder index. The selection of precoder rank should also be included in the precoder selection algorithm. The feedback rate may vary from a short-term feedback once every coherent time interval to a long-term feedback once every several coherent time intervals.
In many systems, the optimal precoders from the codebook for two adjacent transmission blocks are close with respect to a proper distance measure in the set of all possible precoders. Here, the adjacent blocks may be considered in time or in frequency, e.g., over the set of tones in OFDM systems since in practical systems the channel does not change abruptly from one transmission block to the adjacent one. Thus, the precoder used in those blocks should be equal if the channel is pretty steady and the codebook resolution is not two high. By increasing the codebook resolution or having a more dynamic channel, the precoders of the adjacent blocks are not equal anymore, yet, they might be close. The closeness between two precoders can be measured based on a proper distance metric in the space of all such precoders.